Computation of compressible non-isothermal power-law model developed Taylor-Galerkin Pressure-corre

Abstract

Keywords: Compressible flow; developed taylor-galerkin/pressure-correction method; non-isothermal flow; non-newtonian; power-law model

1. Introduction

Non-Newtonian fluids constitute an important class in fluid dynamics, with relevance spanning theoretical understanding and practical applications across multiple industries. The non-constant viscosity of these materials, which includes sauces, blood and polymeric solutions is responsible for flow behaviors that are drastically more complex than the behavior of Newtonian fluids [1-4]. Such complexity is particularly relevant to engineering, biomedical, and environmental fields, which can benefit from more accurate representations of flu-

id dynamics and thereby improve their design processes and operational performance. Thus, the study of non-Newtonian fluids remains an area of considerable research interest. As another example, [5,6] investigate atomization processes for non-Newtonian fluids and [7–10] consider the gas bubbles behaviors within a non-Newtonian liquid matrix. In recent years, there has been growing scholarly interest in the study of Casson fluids. The Casson fluid belongs to a class of non-Newtonian fluids with generalized Newtonian behavior usually associated with the presence of a yield stress, and in recent times it has received increased attention from scholars due to its particular

Corresponding Author E-mail Adress: anasm.ali@uokufa.edu.iq *

Submitted: 17 February 2025 ; Accepted: 31 May 2025 This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University, İstanbul, Türkiye This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

relevance for many biomedical and industrial applications (cf., e.g., [11-20]).

the power-law model for non-Newtonian fluids, has not yet been examined.

The behavior of non-isothermal, non-Newtonian fluids play a pivotal role in a wide range of industrial and engineering applications, such as polymer processing, geothermal flows, food manufacturing, and biomedical systems. In these contexts, the interaction between momentum and thermal transport mechanisms is strongly influenced by the fluid’s rheological properties, particularly when nonlinear viscosity effects are present. While significant research has been conducted on the heat transfer characteristics of such fluids, especially through numerical modeling [21-26], most existing work assumes isothermal or incompressible flow conditions, which simplifies the thermodynamic and constitutive complexities of real systems. One of the most successful, widely applied, and accurate numerical solution methods for these important fluid problems, namely Taylor-Galerkin/Pressure-Correction (TG/PC) is primarily limited to the finite element method (FEM). This approach, introduced by Townsend and Webster [27], has been extensively used to solve a variety of fluid-related problems, primarily at isothermal conditions. For examples of such research, Tamaddon‐Jahromi, et. al, [28] used Taylor–Galerkin algorithms for solving model convection–diffusion problems for simulating more complex non-Newtonian flows. Carew, et. al, are solved a viscoelastic flow using TG/ PC method that incorporates consistent Petrov-Galerkin streamline upwinding within the discretisation of the constitutive equations [29]. A semi-implicit TG/PC was developed to investigate problems that manifest free surfaces associated with the incompressible Newtonian fluids by Ngamaramvaranggul and Webster [30]. [31] TG/ PC method was used for solving die-swell flow for viscoelastic and viscoelastoplastic fluids by Al-Muslimawi. Sharhan and Al-Muslimawi [32] had examine the flow of inelastic fluids with various shear properties by TG/PC algorithm. Al-Haboobi, et. al used TG/PC for first time to study a Newtonian isothermal flow around bluff body for incompressible [33] and compressible fluid [34].

The current work is based on this framework, but extends the earlier works by providing, for the first time, a generalized extension of the DTG/PC method to model non-isothermal, compressible, inelastic fluids governed by an Ostwald-de Waele power-law fluid law [36]. This model is able to capture both shear-thinning and shear-thickening behavior by varying the power-law index (n) and consistency coefficient (m). With the Tait equation accounting for compressibility and the energy equation allowing for thermal coupling, a fully coupled, thermodynamically consistent model is achieved.

A major limitation in the open literature is the lack of high-fidelity numerical methods capable of simulating compressible, non-isothermal, non-Newtonian fluids, particularly those described by the power-law model. Existing studies have primarily addressed either incompressible non-Newtonian flows or compressible Newtonian flows. Challenges arise from the nonlinear coupling among temperature, viscosity, pressure, and density, and from the need to simultaneously solve the energy equation and a pressure-density relation, such as the Tait equation of state.

To fill this research gap, Al-Haboobi and Al-Muslimawi proposed in 2023 a new method for non-isothermal compressible Newtonian fluids, namely the developed Taylor-Galerkin/Pressure-Correction (DTG/PC) method [35]. It solves the energy equation using a twostep Lax-Wendroff scheme and applies a pressure correction for compressibility based on the Tait equation of state. Its application to more complex constitutive laws, specifically those described by

To validate the robustness of the proposed methodology, we conducted a detailed numerical investigation in an axisymmetric cylindrical channel. The effects of key physical parameters (n), (m), (k), Pr namely the power-law index, the consistency parameter, the thermal conductivity, and the Prandtl number on important flow variables (axial velocity (v), pressure (p), density (ρ), temperature (T), shear stress ( x rz ), and normal stress ( x zz ) are analyzed. The originality of this study lies in extending the DTG/PC framework to compressible, non-isothermal power-law fluids, which has not been reported in the literature. The method captures the complex thermodynamic-rheological coupling and produces stable, high-fidelity evolution. This gap is addressed by providing a robust numerical infrastructure for simulating complex flows essential to diverse scientific and industrial applications.

2. Mathematical description

The continuity, momentum, and energy equations that govern non-isothermal, compressible, inelastic, flow are expressed in two-dimensional cylindrical coordinates (r, z) respectively, as [34, 35]:

2 2ev ^ + v.d h v l = d ^ 2n s d h - 3 d ^ n s d.v h - dp. 2t

2v r 1 2p 1 2 2 lE ;n b - d $ v + 2 -t + 2r t 2r s 3 2r 2v r 2v z 1 2 1 2 n s 2v r v r l lE + t r b ;n b +t + , 2z s 2z 2r 2r r

2v z 1 2p 1 2 2 lE ;n b - d $ v + 2 -t + 2r t 2z s 3 2z n 2v z 2v r 1 2 1 s 2v r 2v z l lE + t r b ;n b +t + + . 2r s 2r 2z 2z 2r

1 2 2T 2T 2T 2T 2 2T b kr l+ bk l + U. + vr + vz D= r 2t 2r 2z 2r 2t 2z 2z (7)

Where v = ^ v r, v z h, p, t and n s are the velocity vector, pressure, density, and solvent viscosity of the fluid, respectively. In the energy equation, the two variables each represent the temperature T , the heat capacity Cp , the thermal conductivity k, and the dissipation function U . The deformation tensor’s Euler rate is defined as follows. d = (dv + (dv) @)/ 2. The solvent viscosity is calculated using a power-law model, which is defined as [32]: n-1 n s = m ^ c h (8)

Where, m is consistency parameter, n is power-law index and c represent the shear rate of simple shear flow defined as [32]: c = 2 P . (9)

Where, II denotes the second invariants of the rate of strain tensor, which can be articulated in an axisymmetric coordinate system as follows [32]: 1 2v r l2 b 2v z l2 a v 2r k 1 b 2v r 2v z l2 1 II = 2 ' b . + + r +2 + 2r 2z 2r 2z

To non-dimensionalize the conservation (of mass, momentum and energy), we have used the following variables: v = V 0 v *, t = t 0 t *, p = 0

ns V0 * L2 t 0 * T - T0 1 * p , T , d d , t = = = ns t , L L DT 0

where V is the characteristic velocity, p is the characteristic 0 pressure, t 0 is the characteristic density, T is the characteristic temperature, L is the characteristic length, and DT is a difference reference temperature. Using the quantities defined above, Equations 1 through 6 can be transformed into the following equations: Conservation of mass

2 2v ^ + v.d h v l = d ^ 2n s d h - 3 d ^ n s d.v h - dp. 2t

2v r 1 1 2p 1 1 2 2 lE ;n b - d $ v + 2 - t Re + 2r t Re 2r s 3 2r 2v r 2v z 1 1 2 1 1 2 n s 2v r v r l lE + t Re r b ;n b + t Re + , 2z s 2z 2r 2r r

2v z 1 1 2p 1 1 2 2 lE ;n b - d $ v + 2 - t Re + 2z t Re 2z s 3 2z n s 2v r 2v z 2v z 2v r 1 1 2 1 1 lE + t Re r b l. ;n b + t Re + + 2r s 2r 2z 2z 2r

In component form: 2T 2T 2T t Re Pr : + vr + vz D 2t 2r 2z 1 2 2T 2 2T b kr l+ bk l + Pr EcU. = r 2t 2z 2z 2r

Where Re, Pr and Ec are non-dimensional values representing the Reynolds number, Prandtl number, and Eckert number, respectively, defined as [35,37]: t0 V0 L ns , ns Pr = k/C , p (V 0 ) 2 Ec = C p Dt Re =

Then, we need to specify an equation of state — a relation between density and pressure — to close the two governing equations (mass and momentum) outlined above. The modified Tait of state equation has been assumed in the form [38]:

Here A and B represent parameters, and denote p 0, t 0 the reference values of pressure and density, respectively. By rearranging and differentiating Equation 18, we obtain:

2p A (p + B) (19) = Aat A - 1 = = c 2(x,t) t 2t 0 (p + B) Where, a = is constant and c(x,t) is the speed of sound. t A0

3. Numerical algorithm

In this section, we discuss the DTG/PC method, which is used to solve the governing equations for compressible non-Newtonian fluids with thermal effects. The algorithm employs the Taylor-Galerkin technique, a fractional-step method that initially semi-discretizes time using a Taylor series expansion, and subsequently applies a pressure-correction approach. This numerical strategy was first introduced by Hawken et al. [27] to address incompressible isothermal flows. Subsequently, the Lax-Wendroff two-step [39] procedure was applied to the preceding equations and to derive the following results. Momentum conservation equation becomes: step1: v n + 1/2

Dt step2: v n + 1 = v n + t Re ( (v n + 1/2, d n + 1/2) - dp n + 1/2),

In step 2 of Equation 20, we approximate pressure p n + 1/2 by using Crank-Nicolson method as [40]: p n + 1/2 = jp n + 1 + (1 + j) p n

Now using Equation 22 we can re - written step 2 of Equation 20 as: v

Dt = v + t Re ^ (v n + 1/2, d n + 1/2) - jdp n + 1 - (1 - j) dp n h . n

The velocity v* introduce to solve Equation 23 in conjugation with the compressibility Equation 11, as: Dt v * = v n + t Re ^ (v n + 1/2, d n + 1/2) - dp n h .

Now by subtracting Equation 24 from Equation 23 we have jDt v n + 1 = v ) - tRe dq n + 1,

Taking the divergence of both sides of Equation 25 together with using Equation 11 Dt n + 1 + d $ (t Re v )) = jDtd 2 q n + 1 Dt

Moreover, by applying the chain rule to Equation 19 and performing difference operations, the density increment can be expressed in terms of the pressure increment, Dt n + 1 1 Dp = 2 . c (x,t) Dt Dt n+1

Substituting Equation 27 in to Equation 26 we get n+1 1 Dp + d $ (t Re v )) = jDtd 2 q n + 1 . 2 c (x,t) Dt

Dt step2: T n + 1 = T n + t RePr (G (v n + 1/2, T n + 1/2), (29) where,

Thus, from Equation (20 step 1), and Equations 24, 25, 27, 28, 29, we can solve the system of compressible non-Newtonian inelastic equations at the following stages: stage1:

Now, these stages can be described in matrix/vector form as follows: stage1:

2t n Re † 2 MDV n + 1/2 = L P n - SV n + 3 DV n - t n Re N (V n) V n + B, Dt

t n Re † 2 MDV * = L P n - SV n + 1/2 + 3 DV n + 1/2 - t n Re N (V n + 1/2) V n + 1/2 + B, Dt

2t n RePr MDT n + 1/2 = -CT n - t n RePr N (T n) V n + B + Pr EcU, Dt

t n RePr MDT n + 1 = -CT n + 1/2 - t n RePr N (T n + 1/2) V n + 1/2 + B + Pr EcU, Dt

where, DV n + 1/2 = V n + 1/2 - V n, DV n + 1 = V n + 1 - V *, DT n + 1/2 = T n + 1/2 - T n, DT n + 1 = T n + 1 - T n, Dp n + 1 = p n + 1 - p n, Dt n + 1 = t n + 1 - t n and DV ) = V ) - V n .

V, T, ρ, and P represent a velocity, energy, density, and pressure vector, respectively. The vector of pressure difference is defined as Qn+1 = Pn+1 - Pn . The notation and structure of the matrix are delineated as follows:

S ij = GdK i, dK j + (dK j) † H = # n dK i:( dK j + (dK j) †) dX X

D ij = [dK i, dK j] = # n dK i $ dK j dX , K ij = [d} i, d} j] = # d} i $ d} j dX s

H i,j = (} i, } j) = # } } j dX, B ij = (} i, d $ (} j K j)) = # } d $ (} j K j) dX X

N (T) ij = [K i, T $ dK j] = # _ | l K i K j T l $ d i K j dX X

N (V) ij = [K i, V $ dK j] = # _ | l K i K j V l $ d i K j dX X

where B , g are a natural boundary vector and given function. We approximate the values of velocity, pressure, and temperature as

follows: v = | i v (t) K i, T = | i T (t) K i and p = | j } j p . K i and } j represent the quadratic and linear shape function respectively. i

4. Specification of the problem and boundary conditions

We investigate the compressible Poiseuille flow (Ps) through a straight, two-dimensional (2D) axisymmetric channel under non-isothermal conditions. We use a highly refined triangular finite-element mesh with a normal junction-configuration, as shown in Figure 1. The mesh comprises 1679 elements, 960 vertices, and 3840 degrees of freedom.

on temperature. Such studies are performed for individual values ^ n = 0.5 to1.5 h of ^ m = 1, 5, 10, 20 h and, to illustrate the range of flow behaviour under varying thermal and rheological conditions.

5. Results and discussions

In the present work, we performed systematic simulations of steady, non-Newtonian thermal flow using the DTG/PC method, an advanced and efficient numerical technique. The main purpose of this study is to investigate the impact of the power-law index (n) and the consistency parameter (m) on flow characteristics and to discuss the effect of the Prandtl number (Pr) and the thermal conductivity (k)

Figure 4. Influence of variation n on velocity at outlet We investigate the effects of varying the consistency coefficient (m) at values of 1, 5, 10, and 20, for both shear-thinning (n=0.5) and shear-thickening (n=1.5) fluids, while keeping all other parameters constant (Re=Pr=1). Figure 5 shows that increasing the consistency

coefficient (m) does not affect the inlet velocity profile. This behavior is a consequence of the dominant influence of the inlet boundary conditions, as illustrated in Figure 2.

Figure 5. Influence of variation m on velocity at inlet At the channel outlet, the effect of varying the consistency coefficient is examined using the results presented in Figure 6. The figure demonstrates that increasing m leads to a pronounced rise in velocity relative to that at the channel inlet. Furthermore, it is observed that the influence of increasing m on velocity is more pronounced for power-law index values, which is n < 1 attributed to the shear-thinning behavior of the fluid. Everything that has been mentioned up until this point is consistent with what was provided in [35, 41].

The influence of the consistency coefficient (m={1,5,10,20}) on pressure and density along the centerline is examined in Figure 8 for power-law index values n=0.5 and n=1.5. As illustrated in the figure, with increasing m both p and ρ also increase in each region. However, this behavior is more pronounced for n > 1 (shear-thickening fluids), since the m dependent behavior on the viscosity scaling ratio dominates in shear-thickening regime compared to shear-thinning one [43].

Figure 6. Influence of variation m on velocity at outlet Figure 7 shows the centerline pressure (p) and density (ρ) profiles for different values of the power-law index (n), illustrating pressure- and density-drop profiles as n and m vary. Here we can see the results of ρ and p distribution for maximum values (n=1.5) These observations are consistent with the known physical behaviour of non-Newtonian fluids. Decreasing n increases the fluid’s viscosity (its resistance to flow). Thus, this results in increased pressure and density at the same flow condition [35, 42].

6. Conclusion

Finally, we explore the impact of n and m on shear stress ( x rz ) and normal stress ( x zz ) as shown in Figures 15, 16, and 17. Figure 15 shows that the magnitudes of the shear and normal stresses increase, consistent with the shear-thickening behavior of the fluid. These results are consistent with those reported previously [48-51]. When m varies, Figures 16 and 17 show the effects on n = 0.5 and n = 1.5, respectively. The trend shows that the magnitudes of x rz and x zz increase with m shear-thickening fluids clearly following this pattern. This phenomenon is caused by the increased viscosity for larger values of m (as noted in [43]).

This study successfully extends the developed Taylor-Galerkin/ Pressure-Correction method to simulate compressible, inelastic, non-Newtonian fluid flows under non-isothermal conditions in cylindrical coordinates. This approach accounts for the full coupling of momentum, energy, and mass transport due to compressibility, and provides a direct solution to the energy equation via a twostep Lax-Wendroff scheme. Numerical methods have been shown, through quantitative analyses, to be robust and accurate. At the channel outlet, increasing the power-law index from n=0.5 (shear-thinning) to n=1.5 (shear-thickening) caused a reduction in flow velocity by up to, indicating 10% increased flow resistance. For shear-thickening fluids, increasing the consistency coefficient led m=1 to m=20 an 35% increase in outlet velocity, while shear-thinning flows showed 10% a smaller increase due to lower viscosity sensitivity. Pressure (p) and density (ρ) profiles at the pipe centerline showed a peak pressure of approximately 400 and 4 at n=1.5 and m=20 for shear-thickening fluids, indicating that both parameters are strongly related to flow resistance. Likewise, shear and normal stresses increased by as much as 50% with increasing n and m, highlighting their effect on the fluid stress state. Thermal findings demonstrate that, particularly in compressible flows, the temperature increases 20% as the Prandtl number and thermal conductivity decrease. The temperature at the channel outlet T=180 was higher in shear-thickening m=20 fluids, demonstrating the strong coupling between viscosity and thermal diffusion. The novelty of this work lies in extending the capabilities of the Taylor-Galerkin/Pressure-Correction scheme developed for power-law fluids to include compressibility and non-isothermal effects, which have not previously been addressed in this context. It is suitable for simulating realistic industrial processes involving polymer melts, geophysical flows, or high-speed thermally driven systems. Possible future work includes extending the framework to three-dimensional domains, transient flow regimes, and viscoelastic fluid models. Further development of tailored turbulence models and aggressive mesh adaptation strategies will improve simulation fidelity for a range of engineering applications.

Copyright permission statement All figures included in this manuscript are original works created by the authors. No copyrighted figures, images, or graphical elements from other sources have been used in this manuscript.

Nomenclature

Rohsenow WM, Hartnett JP, Cho YI. Handbook of Heat Transfer. Volume 3. New York: McGraw-Hill; 1998. Wu WT, Massoudi M. Recent advances in mechanics of non-Newtonian fluids. Fluids 2020;5(1). https://doi. org/10.3390/fluids5010010. Sadiq N, Jawad M, Khalid F, Jahan S, Hassan AM. Comparative analysis of non-Newtonian and Newtonian fluid flow with dual slip in the presence of motile microorganisms and nanoparticles. BioNanoScience 2024;14(2):1504-1519. https://doi.org/10.1007/s12668-023-01284-x Gupta S, Sharma PK, Kumar S, Tiwari CM. Influence of buoyancy forces in MHD non-Newtonian convective nanofluid utilizing Buongiorno’s model induced by 3D exponential sheet. J Therm Eng 2024;10(5):1107-1119.https://doi. org/10.14744/thermal.0000854 Mansour A, Chigier N. Air-blast atomization of non-Newtonian liquids. J Non-Newtonian Fluid Mech 1995;58(2):161194. https://doi.org/10.1016/0377-0257(95)01356-Z Shin JH, Lee I, Kim H, Koo J. Spray atomization and structure of supersonic liquid jet with various viscosities of non-Newtonian fluids. Open J Fluid Dyn 2012;2(4):297-304.https:// doi.org/10.4236/ojfd.2012.24A036 Mansour MH, Kawahara A, Sadatomi M. Experimental investigation of gas-non-Newtonian liquid two-phase flows from T-junction mixer in rectangular microchannel. Int J Multiph Flow 2015;72:263-274.https://doi.org/10.1016/j.ijmultiphaseflow.2015.02.019

Santoso A, Goto D, Takehira T, Aslam A, Kawahara A, Sadatomi M. Non-Newtonian two-phase flow characteristics across sudden expansion in horizontal rectangular minichannel. World J Mech 2016;6(8):257-272.https://doi.org/10.4236/ wjm.2016.68021 Gkotsis PK, Evgenidis SP, Karapantsios TD. Influence of Newtonian and non-Newtonian fluid behaviour on void fraction and bubble size for a gas-liquid flow of sub-millimeter bubbles at low void fractions. Exp Therm Fluid Sci 2019;109:109875. https://doi.org/10.1016/j.expthermflusci.2019.109912 Sepulveda J, Montillet A, Della Valle D, Loisel C, Riaublanc A. Deformation of gas-liquid interfaces in a non-Newtonian fluid at high throughputs inside a microfluidic device and effect of an expansion on bubble breakup mechanisms. Chem Eng Sci 2020;213:115413.https://doi.org/10.1016/j.ces.2019.115377 Shateri A, Mirzagoli Ganji A, Jalili P, Jalili B, Domiri Ganji D. Utilizing python for numerical analysis of bioconvection in magnetized Casson-Maxwell nanofluid systems with gyrotactic microorganisms: An investigation of dominant factors. Results Eng 2025;25:1-11.https://doi.org/10.1016/j. rineng.2024.103760 Azar AA, Jalili P, Jalili B, Ganji DD. The comprehensive analysis of magnetohydrodynamic Casson fluid flow with rectangular porous medium through expanding/contracting channel. Multidiscip Model Mater Struct 2025;21(1):68-97. https://doi.org/10.1108/MMMS-07-2024-0179 Mahboobtosi M, Jalili B, Shateri A, Jalili P, Domiri Ganji D. Heat transfer characteristics in the squeezing flow of Casson fluid between circular plates: A comprehensive study. Adv Mech Eng 2024;16(10):1-11.https://doi. org/10.1177/16878132241290942 Nadalinia Chari F, Jalili P, Jalili B, Domiri Ganji D. MHD Casson non-Newtonian fluid flow in a channel with expanding/ contracting porous walls in the presence of thermal radiation. Proc Inst Mech Eng Part E J Process Mech Eng 2024;1-11. https://doi.org/10.1177/09544089241241459 Roshani H, Jalili P, Jalili B, Ahmad I, Hendy AS, Ali MR, Ganji DD. The effect of magnetic field on the heat transfer in the porous medium octagonal cavity with Cassini oval barriers. Case Stud Therm Eng 2024;56:1-11.https://doi.org/10.1016/j. csite.2024.104194 Jalili B, Sadinezhad Fard M, Khan Y, Jalili P, Ganji DD. A novel technique to analyze the fractional model of Williamson and Casson non-Newtonian boundary layer flow. Multidiscip Model Mater Struct 2024;20(2):279-294.https://doi. org/10.1108/MMMS-09-2023-0309 Jalili B, Rezaeian A, Jalili P, Domiri Ganji D, Khan Y. Squeezing flow of Casson fluid between two circular plates under the impact of solar radiation. ZAMM Z Angew Math Mech 2023;103(9):1-11.https://doi.org/10.1002/zamm.202200455 Sadinezhad Fard M, Torabiyan A, Jalili P, Jalili B, Domiri Ganji D. Investigating the magnetohydrodynamics non-Newtonian fluid movement on a tensile plate affected by variable

thickness with Dufour and Soret effects: Akbari Ganji and finite element methods. Int J Electrochem Sci 2024;19(8):1-11. https://doi.org/10.1016/j.ijoes.2024.100701 Ozsahin DU, Jalili B, Asadi Z, Shateri A, Jalili P, Domiri Ganji D, Ahmad H, Nofal TA. Investigation of turbine cooling using semi-analytical methods in non-Newtonian fluid flow with porous wall. Case Stud Therm Eng 2024;53:1-11.https://doi. org/10.1016/j.csite.2023.103808 Makkar V, Poply V. Impact of outer velocity on flow, heat, and mass transfer of Casson nanofluid over a non-linear stretching sheet. J Therm Eng 2021;7(6):1353-1365.https:// doi.org/10.18186/thermal.859221 Ramachandra Prasad V, Abdul Gaffar S, Keshava Reddy E, Anwar Bg O. Numerical study of non-Newtonian Jeffreys fluid from a permeable horizontal isothermal cylinder in non-Darcy porous medium. J Braz Soc Mech Sci Eng 2015;37(6):17651783.https://doi.org/10.1007/s40430-014-0301-5 Sarojamma G, Sreelakshmi K, Animasaun IL. Numerical study of nonlinear thermal radiative heat transfer in a non-Darcy chemically reactive Casson fluid flow. SN Appl Sci 2019;1(10):1-11.https://doi.org/10.1007/s42452-019-1159-z Chang A, Sun HG, Vafai K, Kosari E. Numerical analysis of flow and forced convection heat transfer of non-Newtonian fluid in a pipe based on fractional constitutive model. Int J Numer Methods Heat Fluid Flow 2021;31(8):2680-2697. https://doi.org/10.1108/HFF-10-2020-0637 Ren X, Xin Z, Liu F. A non-Newtonian thermal lattice Boltzmann method for simulation of Rayleigh-Bénard convection of power-law fluids. AIP Adv 2023;13(11):1-11.https://doi. org/10.1063/5.0168404 Chtaibi K, Hasnaoui M, Ben Hamed H, Dahani Y, Amahmid A. Numerical simulations of the Lorentz force effect on thermal convection in an inclined square cavity filled with a non-Newtonian fluid. Lect Notes Mech Eng 2024:196-206. https://doi.org/10.1007/978-3-031-43934-6_21 Vamsi BVSR, Raveendiran P, Sastry MRC. Experimental and numerical study of fluid flow and heat transfer in the impinging of inline round jets. J Therm Eng 2025;11(1):270-289. https://doi.org/10.14744/thermal.0000918 Hawken DM, Tamaddon-Jahromi HR, Townsend P, Webster MF. A Taylor-Galerkin-based algorithm for viscous incompressible flow. Int J Numer Methods Fluids 1990;10(3):327351.https://doi.org/10.1002/fld.1650100307 Tamaddon-Jahromi HR, Ding D, Webster MF, Townsend P. A Taylor-Galerkin finite element method for non-Newtonian flows. Int J Numer Methods Eng 1992;34(3):741-757.https:// doi.org/10.1002/nme.1620340304 Carew EOA, Townsend P, Webster MF. A Taylor-Petrov-Galerkin algorithm for viscoelastic flow. J Non-Newtonian Fluid Mech 1993;50(2-3):253-287.https://doi.org/10.1016/03770257(93)80034-9

Ngamaramvaranggul V, Webster MF. Computation of free surface flows with a Taylor-Galerkin/Pressure-Correction algorithm. Int J Numer Methods Fluids 2000;33(7):993-1026.https://doi.org/10.1002/10970363(20000815)33:7<993::AID-FLD40>3.0.CO;2-A Al-Muslimawi AH. Theoretical and numerical studies of die swell flow. Korea Aust Rheol J 2016;28(3):229-236.https://doi. org/10.1007/s13367-016-0023-6 Sharhan A, Al-Muslimawi A. Inelastic solution for power law fluid with Taylor-Galerkin/Pressure-Correction finite element method: Axisymmetric contraction flows. J Appl Fluid Mech 2023;16(12):2411-2423.https://doi.org/10.47176/ jafm.16.12.1982 Al-Haboobi A, Al-Juaifri GA, Al-Muslimawi AH. Numerical study of Newtonian laminar flow around circular and square cylinders. Results Control Optim 2024;14:1-11.https://doi. org/10.1016/j.rico.2023.100328 Al-Haboobi A, Fadhel IA, Sharhan AA, Al-Muslimawi AH. Computational simulation of compressible Newtonian flow past circular and square bluff bodies. Ocean Syst Eng 2024;14(4):315-330.https://doi.org/10.12989/ ose.2024.14.4.315 Al-Haboobi A, Al-Muslimawi AH. Novel algorithm for compressible Newtonian axisymmetric thermal flow. Int J Mod Phys C 2024;35(3):1-11.https://doi.org/10.1142/ S0129183124500256 Shapovalov VM. On the applicability of the Ostwald-De Waele model in solving applied problems. J Eng Phys Thermophys 2017;90(5):1213-1218.https://doi.org/10.1007/ s10891-017-1676-9 Al-Haboobi A, Al-Muslimawi AH. A new algorithm for solving thermal Newtonian flow in axisymmetric straight channel. Basrah J Sci 2023;41(3):399-418.https://doi.org/10.29072/ basjs.20230301 Dymond JH, Malhotra R. The Tait equation: 100 years on. Int J Thermophys 1988;9(6):941-951.https://doi.org/10.1007/ BF01133262 Heuz T. Lax-Wendroff schemes for elastic-plastic solids. J Comput Phys 2019;396:89-105.https://doi.org/10.1016/j. jcp.2019.06.050 Crank J, Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Adv Comput Math 1996;6(1):207-226. https://doi.org/10.1007/BF02127704 Westermaier S, Kowalczyk W, et al. Implementation of non-Newtonian fluid properties for compressible multiphase flows in OpenFOAM. Open J Fluid Dyn 2020;10(02):135. https://doi.org/10.4236/ojfd.2020.102009 Ikoku CU, Ramey Jr HJ. Transient flow of non-Newtonian power-law fluids in porous media. Soc Pet Eng J 1979;19(03):164-174.https://doi.org/10.2118/7139-PA

Pang B, Wang S, Chen W, Hassan M, Lu H. Effects of flow behavior index and consistency coefficient on hydrodynamics of power-law fluids and particles in fluidized beds. Powder Technol 2020;366:249-260.https://doi.org/10.1016/j.powtec.2020.01.061 Gooch JW, editor. Power law (Ostwald-DeWaele model). In: Springer Handbook of Experimental Fluid Mechanics. Springer New York; 2007. p. 781.https://doi.org/10.1007/9780-387-30160-0_9193 Koponen AI. The effect of consistency on the shear rheology of aqueous suspensions of cellulose micro- and nanofibrils: a review. Cellulose 2020;27(4):1879-1897.https://doi. org/10.1007/s10570-019-02908-w Prasad D, Subrahmanyam SV, Panda SS. Thermal, squeezing and compressibility effects in lubrication of asymmetric rollers. Tribol Ind 2014;36(3):244–258. Dak M, Verma RC, Jaaffrey SNA. Rheological properties of tomato concentrate. Int J Food Eng 2008;4(7):1-11.https:// doi.org/10.2202/1556-3758.1470 Tripathi S. Effects of power law fluid characteristics on core-annular flow in a horizontal pipe. Lect Notes Mech Eng 2024:77-84.https://doi.org/10.1007/978-981-99-6074-3_8 Keslerov R, Kozel K. Numerical modelling of incompressible flows for Newtonian and non-Newtonian fluids. Math Comput Simul 2010;80(8):1783-1794.https://doi.org/10.1016/j. matcom.2009.12.005 Jeong SW, Leroueil S, Locat J. Applicability of power law for describing the rheology of soils of different origins and characteristics. Can Geotech J 2009;46(9):1011-1023.https://doi. org/10.1139/T09-031 Siddiqui AM, Ahmed M, Ghori QK. Couette and Poiseuille flows for non-Newtonian fluids. Int J Nonlinear Sci Numer Simul 2006;7(1):15-26.https://doi.org/10.1515/IJNSNS.2006.7.1.15

Share and Cite